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Robin’s theorem, primes, and a new elementary reformulation of the Riemann hypothesis. (English) Zbl 1235.11082

There are equivalent formulations of the Riemann hypothesis (RH) by G. Robin (1984, using Euler-constant \(\gamma\)) and J. C. Lagarias [Am. Math. Mon. 109, No. 6, 534–543 (2002; Zbl 1098.11005), using harmonic numbers). The authors give another elementary one, using Gronwall’s function \(G(n)= {\sigma(n)\over n\log\log n}\) \((n> 1)\): RH is true if and only if \(n= 4\) is the only composite number with the two properties:
(i) \(G(n)\geq G({n\over p})\) for every prime factor \(p\) of \(n\);
(ii) \(G(n)\geq G(an)\) for every positive integer \(a\).
The proof is elementary and uses the results of Gronwall and Robin.

MSC:

11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11N64 Other results on the distribution of values or the characterization of arithmetic functions
11Y55 Calculation of integer sequences

Citations:

Zbl 1098.11005
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Online Encyclopedia of Integer Sequences:

Superabundant [or super-abundant] numbers: n such that sigma(n)/n > sigma(m)/m for all m < n, sigma(n) being A000203(n), the sum of the divisors of n.
a(n) = floor( exp(gamma) n log log n ) - sigma(n), where gamma is Euler’s constant (A001620) and sigma(n) is sum of divisors of n (A000203).
a(n) = floor( exp(gamma) n log log n ), where gamma is Euler’s constant (A001620).
Positive integers such that sigma(n) >= exp(gamma) * n * log(log(n)).
Values of n for which sigma(n) < e^gamma * n * log(log(n)).
Continued fraction for e^gamma.
Superabundant numbers (A004394) satisfying the reverse of Robin’s inequality (A091901).
Non-superabundant numbers satisfying the reverse of Robin’s inequality (A091901).
Smallest prime factor p of r = A067698(n) such that sigma(r/p)/((r/p)*log(log(r/p))) > sigma(r)/(r*log(log(r))), where sigma(k) = sum of divisors of k; or 1 if no such p.
GA1 numbers: composite m with G(m) >= G(m/p) for all prime factors p of m, where G(k) = sigma(k)/(k*log(log(k))) and sigma(k) = sum of divisors of k.
GA2 numbers: n with G(n) >= G(a*n) for all integers a > 0, where G(k) = sigma(k)/(k*log(log(k))) and sigma(k) = sum of divisors of k.
Proper GA1 numbers: terms of A197638 with at least three prime divisors counted with multiplicity.
Number of GA1 numbers A197638 with n >= 3 prime factors counted with multiplicity.
a(1) = 0, and for n > 1, a(n) = 1 if G(A329886(n)) >= G(A329886(floor(n/2))), otherwise 0, where G(n) = sigma(n) / (n*log(log(n))), where sigma is the sum of the divisors.